Some T2 spaces must have a small dense?

Question

If a Hausdorff space $\ X\ $ admits a dense subset $ A \hookrightarrow X\ $ such that

$$|X|^{|A|}\ =\ |X|$$

then indeed $$|X| \leq |\text{End}_{\text{Top}}(X)| \leq |X|^{|A|}\ = \ |X|.$$

It is the case of $\mathbb{Q} \hookrightarrow \mathbb{R}$. Thus, if there is a small enough dense subspace, there are not so many endomorphisms.

Is the converse true?

Namely, suppose $|\text{End}_{\text{Top}}(X)| = \ |X|.$ Is it true that there exists a dense subset $A$ such that $|X|^{|A|}\ =\ |X|$?

UPDATE: Solved on MO.

You can construct group topologies on sets where the only endomorphism (homeomorphism of $X$ with itself, in my definition) is a translation. Make such a space on a countable set and voila. No time to write it out now. Maybe later. — Henno Brandsma, Nov 23 '17 at 22:38
@HennoBrandsma $\text{End}_{\text{Top}}(X)$ as in endomorphisms in the $\text{Top}$ category, i.e. a continuous $X\to X$ function. There's nothing wrong with the notation. Anyway this is an interesting question. Seems to be hard though. — freakish, Nov 24 '17 at 09:41
@freakish so $\text{Hom}(X,X)$ essentially, in category theory terms. — Henno Brandsma, Nov 24 '17 at 10:08
I think that the condition $|C(X,X)| = |X|$ is quite rare. We always have $|C(X,X)| \ge |X|$. — Henno Brandsma, Nov 26 '17 at 16:08

score 1 · Accepted Answer · answered May 08 '19 at 09:53

(To kick it from the unanswered queue)

As mentioned by the OP, this is solved on MO: Some T2 spaces must have a small dense?

@KeithKearnes 's solution:

Is the converse true?

No.

The paper Constructions and Applications of Rigid Spaces, I, Advances in Mathematics 29 (1978), 89-130, by Kannan and Rajagopalan, describes a countaby infinite Hausdorff space $X$ such that the only continuous maps $f\colon X\to X$ are the constant maps and the identity map. Such a space satisfies $|\text{End}_{\text{Top}}(X)| = \omega = \ |X|.$

If $A\subseteq X$ is a subset such that $|X|^{|A|}=|X|=\omega$, then $A$ is forced to be finite, and therefore $A$ cannot be dense.

Some T2 spaces must have a small dense?

1 Answers1